The Gibbs–Thomson effect, in common physics usage, refers to variations in vapor pressure or chemical potential across a curved surface or interface. The existence of a positive interfacial energy will increase the energy required to form small particles with high curvature, and these particles will exhibit an increased vapor pressure. See Ostwald–Freundlich equation. More specifically, the Gibbs–Thomson effect refers to the observation that small crystals are in equilibrium with their liquid melt at a lower temperature than large crystals. In cases of confined geometry, such as liquids contained within porous media, this leads to a depression in the freezing point / melting point that is inversely proportional to the pore size, as given by the Gibbs–Thomson equation.

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The technique is closely related to using gas adsorption to measure pore sizes, but uses the Gibbs–Thomson equation rather than the Kelvin equation. They are both particular cases of the Gibbs Equations of Josiah Willard Gibbs: the Kelvin equation is the constant temperature case, and the Gibbs–Thomson equation is the constant pressure case.[1] This behaviour is closely related to the capillary effect and both are due to the change in bulk free energy caused by the curvature of an interfacial surface under tension.[2][3] The original equation only applies to isolated particles, but with the addition of surface interaction terms (usually expressed in terms of the contact wetting angle) can be modified to apply to liquids and their crystals in porous media. As such it has given rise to various related techniques for measuring pore size distributions. (See Thermoporometry and Cryoporometry.) The Gibbs–Thomson effect lowers both melting and freezing point, and also raises boiling point. However, simple cooling of an all-liquid sample usually leads to a state of non-equilibrium super cooling and only eventual non-equilibrium freezing. To obtain a measurement of the equilibrium freezing event, it is necessary to first cool enough to freeze a sample with excess liquid outside the pores, then warm the sample until the liquid in the pores is all melted, but the bulk material is still frozen. Then, on re-cooling the equilibrium freezing event can be measured, as the external ice will then grow into the pores.[4][5] This is in effect an "ice intrusion" measurement (cf.mercury intrusion), and as such in part may provide information on pore throat properties. The melting event can be expected to provide more accurate information on the pore body.

Very similar equations may be applied to the growth and melting of crystals in the confined geometry of porous systems. However the geometry term for the crystal-liquid interface may be different, and there may be additional surface energy terms to consider, which can be written as a wetting angle term cos⁡ϕ{\displaystyle \cos \phi \,}. The angle is usually considered to be near 180°. In cylindrical pores there is some evidence that the freezing interface may be spherical, while the melting interface may be cylindrical, based on preliminary measurements for the measured ratio for ΔTf/ΔTm{\displaystyle \Delta \,T_{f}/\Delta \,T_{m}} in cylindrical pores.[7]

Thus for a spherical interface between a non-wetting crystal and its own liquid, in an infinite cylindrical pore of diameter x{\displaystyle x}, the structural melting point depression is given by:[8]

As early as 1886, Robert von Helmholtz (son of the German physicist Hermann von Helmholtz) had observed that finely dispersed liquids have a higher vapor pressure.[11] By 1906, the German physical chemist Friedrich Wilhelm Küster (1861–1917) had predicted that since the vapor pressure of a finely pulverized volatile solid is greater than the vapor pressure of the bulk solid, then the melting point of the fine powder should be lower than that of the bulk solid.[12] Investigators such as the Russian physical chemists Pavel Nikolaevich Pavlov (or Pawlow (in German), 1872–1953) and Peter Petrovich von Weymarn (1879–1935), among others, searched for and eventually observed such melting point depression.[13] By 1932, Czech investigator Paul Kubelka (1900–1956) had observed that the melting point of iodine in activated charcoal is depressed as much as 100°C.[14] Investigators recognized that the melting point depression occurred when the change in surface energy was significant compared to the latent heat of the phase transition, which condition obtained in the case of very small particles.[15]

Neither Josiah Willard Gibbs nor William Thomson (Lord Kelvin) derived the Gibbs–Thomson equation.[16] Also, although many sources claim that British physicist J. J. Thomson derived the Gibbs–Thomson equation in 1888, he did not.[17] Early in the 20th century, investigators derived precursors of the Gibbs–Thomson equation.[18] However, in 1920, the Gibbs–Thomson equation was first derived in its modern form by two researchers working independently: Friedrich Meissner, a student of the Estonian-German physical chemist Gustav Tammann, and Ernst Rie (1896–1921), an Austrian physicist at the University of Vienna.[19][20] These early investigators did not call the relation the "Gibbs–Thomson" equation. That name was in use by 1910 or earlier;[21] it originally referred to equations concerning the adsorption of solutes by interfaces between two phases — equations that Gibbs and then J. J. Thomson derived.[22] Hence, in the name "Gibbs–Thomson" equation, "Thomson" refers to J. J. Thomson, not William Thomson (Lord Kelvin).

In 1871, William Thomson published an equation describing capillary action and relating the curvature of a liquid-vapor interface to the vapor pressure:[23]

could be derived from Kelvin's equation.[24][25] The Gibbs–Thomson equation can then be derived from the Ostwald-Freundlich equation via a simple substitution using the integrated form of the Clausius–Clapeyron relation:[26]

The Gibbs–Thomson equation can also be derived directly from Gibbs' equation for the energy of an interface between phases.[27][28]

It should be mentioned that in the literature, there is still not agreement about the specific equation to which the name "Gibbs–Thomson equation" refers. For example, in the case of some authors, it's another name for the "Ostwald-Freundlich equation"[29]—which, in turn, is often called the "Kelvin equation"—whereas in the case of other authors, the "Gibbs–Thomson relation" is the Gibbs free energy that's required to expand the interface,[30] and so forth.

^As early as 1906, the Austrian mineralogist Cornelio August Doelter [ De ] (1850-1930) had attempted to determine the melting points of various minerals via a microscrope and had observed that finely powdered silicates melted over a range of as much as 100°C. See pp. 618-619 of: Doelter. C (17 August 1906) "Bestimmung der Schmelzpunkte vermittelst der optischen Methode" (Determination of melting points by means of an optical method), Zeitschrift für Elektrochemie und angewandte physikalische Chemie, 12 (33) : 617-621. From p. 618: " … wir erkennen, dass zwischen Beginn der Schmelzung und diesem Punkt bei manchen Silikaten ein erheblicher Temperaturunterschied — bis 100° — liegen kann, … " ( … we discern that between the beginning of melting and this point [i.e., at which molten droplets join together] there can lie, in the case of some silicates, a considerable difference in temperature — up to 100°C … )

^See: Kubelka, Paul (July 1932) "Über den Schmelzpunkt in sehr engen Capillaren" (On the melting point in very narrow capillaries), Zeitschrift für Elektrochemie und angewandte physikalische Chemie (Journal for Electrochemistry and Applied Physical Chemistry), 38 (8a) : 611–614. Available on-line in English translation at: National Research Council Canada. From page 614: "Tests which will be reported in detail by the author elsewhere enable us to prove ... that iodine in activated charcoal is still liquid at room temperature, i.e., approximately 100° below the melting point."

^Sir Joseph John Thomson derived Kelvin's equation (page 163) and the depression of the melting point of ice by pressure (page 258), but he did not derive the Gibbs–Thomson equation. However, on pages 251–252, Thomson considered the effects of temperature and surface tension on the solubility of salts in spherical droplets, and he obtained an equation for that phenomenon which has a form similar to that of the Gibbs–Thomson equation. See: Thomson, J.J., Applications of dynamics to physics and chemistry (London, England: Macmillan and Co., 1888).

A.B. Macallum (October 7, 1910) "Surface tension in relation to cellular processes,"Science, 32 (823) : 449–458. After explaining the Gibbs–Thomson principle (and its origin) on page 455, he uses the term "Gibbs–Thomson principle" on page 457: "This determination is based on the deduction from the Gibbs–Thomson principle that, where in a cell an inorganic element or compound is concentrated, the surface tension at the point is lower than it is elsewhere in the cell."

Duncan A. MacInnes and Leon Adler (1919) "Hydrogen overvoltage", Journal of the American Chemical Society, 41 (2) : 194–207. "By the Gibbs–Thompson law, substances that lower the surface tension are those which are adsorbed."

Shanti Swarup Bhatnagar and Dasharath Lal Shrivastava (1924) "The optical inactivity of the active sugars in the adsorbed state – a contribution to the chemical theory of adsorption. I", Journal of Physical Chemistry, 28 (7) : 730–743. From page 730: "There are at present three well-known theories regarding the mechanism of protective action of colloids: (1) That the protecting agent concentrates at the interface of the colloid particles and the dispersion medium according to [the] Gibbs–Thomson law, ..."

Ashutosh Ganguli (1930) "On the adsorption of gases by solids", Journal of Physical Chemistry, 34 (3) : 665–668. From page 665: "The intimate connection of adsorption with surface tension was shown long before by Gibbs, subsequently known as the Gibbs–Thomson equation."

^Frederick George Donnan and Arthur Erich Haas, ed.s, A Commentary on the Scientific Writings of J. Willard Gibbs, vol. 1 (New Haven, Connecticut: Yale University Press, 1936), page 544. In 1878, Gibbs published an equation concerning the adsorption of a solute by an interface between two phases, and in 1888, J.J. Thomson published an equation concerning the same phenomenon, which he'd derived via a different method but which superficially resembled Gibbs' result. Apparently both equations were eventually known as "the Gibbs–Thomson equation". From page 544: "There is a rather prevalent impression that the two equations are the same, but that is not so; and both on grounds of priority and because of the wider scope of Gibbs' result, there is no justification for the use of the name "Gibbs–Thomson equation" which one sometimes meets in the literature, although it is doubtless true that Thomson's work was independently carried out."